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Theorem brabga 4279
Description: The law of concretion for a binary relation. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypotheses
Ref Expression
opelopabga.1  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
brabga.2  |-  R  =  { <. x ,  y
>.  |  ph }
Assertion
Ref Expression
brabga  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Distinct variable groups:    x, y, A    x, B, y    ps, x, y
Allowed substitution hints:    ph( x, y)    R( x, y)    V( x, y)    W( x, y)

Proof of Theorem brabga
StepHypRef Expression
1 df-br 4024 . . 3  |-  ( A R B  <->  <. A ,  B >.  e.  R )
2 brabga.2 . . . 4  |-  R  =  { <. x ,  y
>.  |  ph }
32eleq2i 2347 . . 3  |-  ( <. A ,  B >.  e.  R  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
41, 3bitri 240 . 2  |-  ( A R B  <->  <. A ,  B >.  e.  { <. x ,  y >.  |  ph } )
5 opelopabga.1 . . 3  |-  ( ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps )
)
65opelopabga 4278 . 2  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( <. A ,  B >.  e.  { <. x ,  y >.  |  ph } 
<->  ps ) )
74, 6syl5bb 248 1  |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A R B  <->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   {copab 4076
This theorem is referenced by:  braba  4282  brabg  4284  epelg  4306  brcog  4850  fmptco  5691  ofrfval  6086  wemaplem1  7261  oemapval  7385  wemapwe  7400  fpwwe2lem2  8254  fpwwelem  8267  clim  11968  rlim  11969  vdwmc  13025  isstruct2  13157  brssc  13691  isfunc  13738  isfull  13784  isfth  13788  ipole  14261  eqgval  14666  frgpuplem  15081  dvdsr  15428  ulmval  19759  hlimi  21767  isumgra  23278  iseupa  23292  isside1  25577  islindf  26694  isuslgra  27501  isusgra  27502
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078
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